Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

Abstract

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian $H:{\mathbb{R}}^n\to \mathbb{R}$ in the regime of low temperature $\epsilon$. We proof the Eyring-Kramers formula for the optimal constant in the Poincaré (PI) and logarithmic Sobolev inequality (LSI) for the associated generator $L=\epsilon \Delta -\nabla H\cdot \nabla$ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 302-351] and of the mean-difference estimate introduced by Chafaï and Malrieu [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 72-96]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to metastable regions corresponding to the local minima scales well in $\epsilon$. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.

Published in at http://dx.doi.org/10.1214/14-AOP908 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Kramers' law: Validity, derivations and generalisations
• Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains

Cited by in corpus (25)

• Generalisation of the Eyring-Kramers transition rate formula to irreversible diffusion processes
• Kramers' law: Validity, derivations and generalisations
• On Generalization Error Bounds of Noisy Gradient Methods for Non-Convex Learning
• Stochastic Modelling of Urban Structure
• Mathematical Analysis of Temperature Accelerated Dynamics
• Accelerating Langevin Sampling with Birth-death
• Sharp spectral asymptotics for non-reversible metastable diffusion processes
• Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation
• Full metastable asymptotic of the Fisher information
• On the Generalised Langevin Equation for Simulated Annealing
• Replica Exchange for Non-Convex Optimization
• Poincaré and log-Sobolev inequalities for mixtures
• Schr{ö}dinger-F{ö}llmer Sampler: Sampling without Ergodicity
• Well-Posedness and Equilibrium Behaviour of Overdamped Dynamic Density Functional Theory
• Gaussian Approximations for Probability Measures on ${\mathbf{R}}^d$
• Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities
• Simulated annealing from continuum to discretization: a convergence analysis via the Eyring--Kramers law
• Sampling Rare Event Energy Landscapes via Birth-Death Augmented Dynamics
• Mass transport in Fokker-Planck equations with tilted periodic potential
• Formulae for the Derivative of the Poincaré Constant of Gibbs Measures
• Passage time of the frog model has a sublinear variance
• An entropic approach for Hamiltonian Monte Carlo: the idealized case
• Convergence Analysis of Schr{ö}dinger-F{ö}llmer Sampler without Convexity
• Higher Order Generalization Error for First Order Discretization of Langevin Diffusion
• Sharp spectral gap of adaptive Langevin dynamics